MLR vs. WLSMV PreviousNext
Mplus Discussion > Structural Equation Modeling >
 Sanjoy posted on Tuesday, May 10, 2005 - 6:49 pm
Dear Professor/s … from earlier MPlus discussions I realize that MLR in effect stands for ML(Full Information) with Huber-White covariance adjustment, which gives us robustness in presence of Non-normality and non-independence of observation … also on page 366 MPlus User’s Guide, u are suggesting MLR as an alternative for WLSMV … I have couple of quick questions in this regard

Q1. My dependent (indicator) variables are categorical, hence non-normal … but ARE THEY SAME in the sense non-normality is being handled in “Sandwich” estimator a.k.a. Huber-White

Q2. Can we use MLR in SEM where we have both measurement model (on multiple categorical indicator) and structural equation system inclusive of covariates (X’s).

Q3. If yes, then can u please suggest me some reference which is kind of counterpart of your (83,84,95,97) articles

Q4. Again (iff we have an yes to Q2.), …now FIML makes difference from Limited information ML (LIML) only when we are estimation some system of equations, I mean at least more than one-equation system (say e.g. ours is a three equation system with endogeneity) … so what numerical method is being used in MPlus when it runs FIML not LIML in order to solve high-order integral

Q5. Now if my understanding about your WLSMV is correct … then it starts with 2 stage least square approach (which is LIML) and later adjusts the covariance matrices using appropriate weight matrix (ur 97 paper which upgraded ur earlier WLS estimator to a more robust WLSMV) … and doing so, we simply circumvent the computational burden of FIML (which sometimes become infeasible) and yet at the same time get a mean –covariance adjusted roust coefficients … then what is the real need for MLR, are we still missing anything substantial in weighted approach

Below is my model
R by R1-R3;
B by B1-B3;
Y on R B X1;
R on B X2;
B on R X3;

R1-R3 and B1-B3 are 5-point categorical, Y is in 1/0 …X’s are the covariate and they do share some common elements

Thanks and regards
 bmuthen posted on Wednesday, May 11, 2005 - 8:23 am
Q1. No, declaring your outcomes as categorical leads to non-linear models. The non-normality adjustments by Huber-White refers to treating the outcomes as continuous.

Q2. Yes.

Q3. See Hu & Bentler in a fairly recent Soc Meth article.

Q4. If the ML estimation requires numerical integration, Mplus offers 3 methods with variations such as adaptive quadrature or not, and Cholesky decomposition (see User's Guide).

Q5. WLSMV is not as efficient as ML, although the loss seems small. ML handles MAR whereas WLSMV cannot given its pairwise variable orientation.
 Sanjoy posted on Wednesday, May 11, 2005 - 2:33 pm
Thank you Prof ...I couldn't find any "Hu & Bentler" article on SEM with Categorical (ordinal) indicator outcome variable ... as per your suggestion, I have looked for it, made a search through “Google”, except their 1999 article on Model Fit index (SEM: AMJ, Vol6 (1), 1-55), which I have already requested from Inter-Library loan, I couldn't find any

And in "Sociological methodology" I could not find any article written by them in particular, instead I found three articles by Prof. Bentler coauthored with some other folks

1.“Assessing the Effect of Model Misspecifications on Parameter Estimates in Structural Equation Models” Ke-Hai Yuan, Linda L. Marshall and Peter M. Bentler, Sociological Methodology, Volume 33, Issue 1, Page 241-265, January 2003

2. “Three Likelihood-Based Methods for Mean and Covariance Structure Analysis with Nonnormal Missing Data” Ke-Hai Yuan; Peter M. Bentler, Sociological Methodology, Vol. 30 (2000), pp. 165-200

3. “Structural Equation Modeling with Robust Covariances” Ke-Hai Yuan; Peter M. Bentler, Sociological Methodology , Vol. 28 (1998), pp. 363-396

and in “Sociological methods & research.” I couldn’t find any article even written by Dr. Bentler (assuming our Library online Journal search engine works ok)

Could you please mention the name of the article?

Using your WLSMV, I got the satisfactory result, however I want to use a ML(Full Information) if at all there is some established statistical theory (like the yours one for WLSMV) which can handle SEM with categorical(ordinal) indicator outcomes along with covariates and I hope, in that case MPlus is able to handle that theory … since all I have is MPlus and one month of time in my hand

Thanks and regards
 bmuthen posted on Wednesday, May 11, 2005 - 4:31 pm
I should have said Yuan & Bentler (2000). Please also see references in Mplus Web Note #2. There is nothing written on ordinal outcomes, only non-normal continuous outcomes.

ML can be quite time consuming if you need many dimensions of integration.
 Sanjoy posted on Wednesday, May 11, 2005 - 8:05 pm
yes professor, I'm going through their 2000 article

... practically speaking apart from your and Prof. Arminger(JASA,92 and his chapter in 95 Handbook) I haven't seen any statistical articles comprehensively dealing with ordinal indicator variables in SEM framework, if I'm not wrong "GLLAMM" can't do that either, I mean not in scenario where latent factors being regressed on other latent factors along with covariates ...

following ur advice I started reading Little's book on missing data(2002 ed.), well if I got them correctly then their chapters on categorical data is primarily concerned about categorical (nominal) rather than categorical(ordinal)one ...I haven't finished their book yet, however this is my first impression

thanks and regards
 Bpnnie posted on Friday, December 16, 2005 - 12:57 pm

In my model, the mediator is categorical while the outcome is continuous. I learned that WLSMV estimator would use probit regression results and ML would report logistic regression results. Which one is better? The default one or logistic results? Would appreciate it !

 Linda K. Muthen posted on Friday, December 16, 2005 - 3:08 pm
Maximum likelihood is a more efficient estimator than weighted least squares. If you can use it to estimate your model, then I would.
 Eva Van de gaer posted on Tuesday, April 24, 2007 - 7:47 am

I have a cross-lagged panel design and I am using Mplus to test the following model:

VARIABLE:NAMES ARE clus g u2-u5 x1-x5;
USEVARIABLES u2-u5 x2-x5;
MODEL: x5 ON x4 x3 x2 u5;
x4 ON x3 x2 u4;
x3 ON x2 u3;
x2 ON u2;
u5 ON u4 x4;
u4 ON u3 x3;
u3 ON u2 x2;
OUTPUT: modindices standardized;

Mplus automatically chooses the WLSMV estimator.

I recently received a comment of a Reviewer to support the use of this estimator. Can you direct me to a reference or an article where the appropiateness of the use of this estimator is explained (in my case/ in this model)? He/she refers to Firth (1992) and McCullagh (1992) who question the use of a sandwich estimator in adjusting covariance matrices.

The same reviewer also asks me to give a rationale for the reporting of the fit indices CFI, TLI and RMSEA. I know that these are (some of) the standard fit indices that Mplus produces but is there a rationale behind the choice of these fit indices above others and if so, can you direct me to a reference where this is explained?

Thank you very much!
 Linda K. Muthen posted on Tuesday, April 24, 2007 - 8:37 am
The following paper which is available on Bengt Muthen's UCLA website studies the WLSMV estimator:

Muthén, B., du Toit, S.H.C. & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Accepted for publication in Psychometrika.

The sandwich estimator is a commonly accepted approach which is widely used.

See Hu & Bentler several years ago in Psycho Methods regarding a variety of fit statistics. See the Yu dissertation on this website for a study of fit statistic behavior for WLSMV.
 Sandra Lyons posted on Monday, May 17, 2010 - 10:05 am
I have SEM output with ordinal indicators thus WLSMV estimates. I need to learn how to interpret and report the estimates and how missing data is handled. Will you please point me to appropriate references. Thank you.
 Linda K. Muthen posted on Tuesday, May 18, 2010 - 9:05 am
With WLSMV, probit regressions are estimated. One good reference is one of the Agresti books on categorical data analysis.

With WLSMV and no covariates, pairwise present is used. This means that each correlation is estimated using all available data. The Little and Rubin book may cover this topic.
 Sandra Lyons posted on Tuesday, May 18, 2010 - 3:54 pm
Thank you, Linda. There are covariates in my model. What does this mean with regard to missing.
 Serge Rijsdijk posted on Thursday, July 01, 2010 - 1:32 am
Dear professors,

In a paper we use the WLSMV estimator because we have a continuous mediator and categorical dependent. I have 2 questions concerning the estimator.

1. I have difficulty finding out whether WLSMV automatically executes the Huber-White correction or whether I should do this manually by providing a weight in the syntax. Could you help me on this?

2. One of the reviewers wants some more information on the estimator and I am looking for a good reference on this. I tried to find the reference mentioned above by Muthen et al. in Psychometrika, but I can't find in anywhere. Do you have (another) suggestion for a good reference describing the ins and outs of WLSMV?

Many thanks,

Serge Rijsdijk
 Linda K. Muthen posted on Thursday, July 01, 2010 - 9:14 am
You can also use maximum likelihood estimation in this case.

1. The standard errors are like Huber-White. MLR provides Huber-White standard errors. With WLSMV you do not need to provide a weight.

2. The Muthen et al paper is on the website under Papers. See also

Muthén, B. & Satorra, A. (1995). Technical aspects of Muthén's LISCOMP approach to estimation of latent variable relations with a comprehensive measurement model. Psychometrika, 60, 489-503.
 ClaudiaBergomi posted on Friday, December 10, 2010 - 8:46 am
Dear Linda,

I am testing a mediation model with dichotomous dependent variable. As I prefer logistic (to probit) regression I took the MLR estimator instead of WLSMV.

As usual, I wish to first calculate a measurement/correlational model and then the mediation models. Unfortunately, MLR does not seem to work for this first model as I get error messages:

*** ERROR in MODEL command
Variances for categorical outcomes can only be specified using
Covariances for categorical variables with other variables are not defined.

Am I doing something wrong or does this mean that I should calculate the measurement model with WLSMV and the mediation models with MLR?
Would it make sense to use these two different estimators for testing the same model (in the sense of same variables)?

Thank you,
 Linda K. Muthen posted on Friday, December 10, 2010 - 9:20 am
Variances of categorical variables are not parameters in cross-sectional models using weighted least squares estimation or maximum likelihood. You should remove specifications of variances for categorical outcomes from the MODEL command.

With weighted least squares estimation and the Theta parametrization, variances can be estimated for multiple group and growth models.
 Averdijk posted on Tuesday, August 23, 2011 - 1:36 am
Dear Dr. Muthen,

I estimated an MSEM model with all binary observed variables (1-(1-1)-1 design; few missing values). I ran the model both using MLR and WLSMV, but get very different results in terms of significance. I realize that MLR uses logit and WLSMV probit, but I expected the significance levels to be similar. When I rerun the direct effects in Stata (xtlogit, fe), these results are much more similar to the MLR results than to WLSMV.
Please find my syntax below. Do you have any suggestions on why the MLR and WLSMV results differ, and whether one is more accurate than the other? Many thanks in advance.

VARIABLE: names are key x1 x2 x3 x4 y;
usevariables are x1 x3 y x4 x2;
categorical = x1 x3 y x4 x2;
missing are all (999);
Cluster = key;
Within are x1 x3 y x4 x2;
ANALYSIS: type = twolevel;
estimator = mlr;
integration = montecarlo;
MODEL: %within%
x3 on x1(a1);
x4 on x1(a2);
y on x3(b1);
y on x4(b2);
y on x2;
x1 on x2;
y on x1;
NEW(indw1 indw2);

Same as MLR, except: estimator = wlsmv;
 Linda K. Muthen posted on Tuesday, August 23, 2011 - 6:20 am
Other than scale differences, the results should be close. Please send the two outputs and your license number to
 Linda K. Muthen posted on Tuesday, August 23, 2011 - 11:58 am
Thanks for sending the outputs. You have missing data. The way missing data are handled is different between WLSMV and MLR. I would use MLR.
 Dan Sass posted on Monday, September 19, 2011 - 8:37 am

I am aware that it is inappropriate to evaluate the change in approximate fit indices (ÄAFI) when doing invariance testing with WLSMV, whereas this is not a problem with ML. Is it appropriate to evaluate the ÄAFI with MLR?

 Linda K. Muthen posted on Monday, September 19, 2011 - 11:12 am
With MLR you must use the scaling correction factor that is provided. See Chi-square Difference Test for MLM and MLR on the website.
 Dan Sass posted on Monday, September 19, 2011 - 11:23 am
I understand that is the case for the change in chi-square, but what about the change in CFI, TLI, RMSEA, and SRMR?
 Linda K. Muthen posted on Tuesday, September 20, 2011 - 10:12 am
I know of no theory related to using the above fit statistics for difference testing.
 Dan Sass posted on Tuesday, September 20, 2011 - 3:07 pm
Perhaps my question was poorly worded.

Based on my understanding, it is statistically inappropriate to evaluate the change in approximate fit indices (e.g., change in CFI = CFI for the measurement invariant model minus the CFI for the configural invariant model) because WLSMV does not allow for a direct comparison between models due to the adjusted chi-squares. Consequently, more emphases should be placed on the change in chi-square using the DIFFTEST procedure than proposed model fit criteria (e.g., Chen, 2007; Cheung & Rensvold, 2002; Meade et al., 2008) using the change in approximate fit indices.

My question is whether the same logic applies to MLR estimation because of the scaling factors, which are likely different for the measurement invariant and configural invariant models. I do not think the scaling factors should influence the approximate fit indices (thus change in approximate fit indices), but I wanted to make sure.

Thanks for your time!!
 Bengt O. Muthen posted on Tuesday, September 20, 2011 - 3:28 pm
I don't know of a theory for comparing fit indices such as CFI even in the case of ML for continuous normal variables. So I don't think it is a matter of DIFFTEST or MLR scaling factors.

MLR scaling factors do play into the computation of CFI because CFI is based on the MLR chi-square which is in turn is affected by the MLR scaling factor.
 Lucio Masserini posted on Tuesday, September 27, 2011 - 1:13 am
Dear Professors,
I'm interested in estimating a two-level structural equation model with Mplus using the MLR estimator. So I would like to read something about this estimator. I found the following bibliography:
Asparouhov & Muthén (2003a). Full-information maximum-likelihood estimation of general two-level latent variable models. Manuscript in preparation.
Asparouhov, T. & Muthén, B. (2003b). Maximum-likelihood estimation in general latent variable modeling. Manuscript in preparation.
Please, can you indicate me where I can find them? Or where I can find something similar?
 Linda K. Muthen posted on Tuesday, September 27, 2011 - 10:09 am
I would recommend looking at Technical Appendix 8 on the website. The MLR estimator is discussed here. See formula 170. Other estimators are described in Technical Appendix 4.
 Minnik Findik posted on Thursday, March 08, 2012 - 11:06 am
Dear Professor/s,

My dependent (indicator) variables are latent (composed of categorical variables). When I run the model, Mplus automatically chooses WLMSV. I can not choose ML since I have more than 4 factors and I have a lot of missing variables.

Hence,I was wondering
1) whether I should choose MLMV or WLSMV?
2) If I choose MLMV (I need to do listwise deletion). I was wondering if I choose WLSMV and do not do listwise deletion, would it just use all the data available? and will that cause a problem??
3) Lastly, if I choose WLSMV will all the regressions be probit coefficients or since the outcome variable is a latent variable will it be linear regression coefficient?
Kind regards.
 Bengt O. Muthen posted on Thursday, March 08, 2012 - 11:15 am
What do you mean by your DVs being latent? Do you mean that you have a factor model for categorical items and the factors are DVs? And you have missing on those categorical factor indicators?
 Minnik Findik posted on Thursday, March 08, 2012 - 11:44 am
I am very sorry to confuse you. I will try to explain it with an example. What I mean is my outcome variable is a latent variable (composed of categorical items).

Use variables are
x81 x115 x97 x31 x32 x33 y1;
Categorical are
x81 x115 x97 x31 x32 x33;
aliena by x81 x115 x97;
alienb by x31 x32 x33;
aliena alienb on y1;
alienb on aliena;

And I have missing values from the items (i.e. in x81 I have 32 missing, in y1 I have 100 missing etc.).

Hence when I do listwise deletion I lose a lot of data (from 13530 to 4000). So firstly, I was wondering which estimator I should be choosing. And I was thinking it would be WLSMV but I am not 100% sure. Secondly, I was wondering if I do not put listwise = deletion will it cause a problem? Lastly,I was wondering if I choose WLSMV will all the regressions be probit coefficients or since the outcome variable is a latent variable will it be linear regression coefficient?

Kindest regards.
 Linda K. Muthen posted on Thursday, March 08, 2012 - 1:43 pm
With a lot of missing data, I would use MLR. I would not use listwise deletion. You have only two factors so this will require two dimensions of intergration which should be okay.

The regression of the factors on the covariates are linear regressions.
 Tihomir Asparouhov posted on Thursday, March 08, 2012 - 1:52 pm
The best estimator in the presence of missing data for you would be the ML estimator. In your model there are only 2 factors so I am not sure how you see more than 4 factors. You should be able to obtain the ML estimates by specifying the appropriate type of numerical integration:

If you have 1-3 factors you can use the default integration=15 (no need of a special command).

If you have 4 factors you can use integration=monte(5000) or integration=10.

For 5 or more factors use integration=monte(5000).

Alternatively you can use multiple imputation routines available in Mplus or the Bayes estimation. See Section 3 in

You should add to the model "y1;" which essentially changes the y1 variable into a dependent variable and thus when y1 is missing the data will not be deleted.

I would not recommend using listwise deletion to deal with the missing data.
 Minnik Findik posted on Thursday, March 08, 2012 - 1:54 pm
Dear Prof. Linda,

thanks a lot for your quick reply. I gave that as an example. In my real model I have 5 factors. That is why I was thinking I wouldn't be able to use MLR hence was thinking to use WLSMV. Do you think that is appropriate? and is it aright to not use listwise deletion in WLSMV? And lastly,in WLSMV the regression of the factors (even they are made up of categorical items)are linear, am I correct?
Thanks a lot for your help,
Kindest regards.
 Bengt O. Muthen posted on Thursday, March 08, 2012 - 2:23 pm
Yes, you don't have to use listwise deletion for WLSMV. The sample statistics of WLSMV uses pairwise present data. And yes, the factor regressions are linear because the DV is continuous.
 Ray Reichenberg posted on Thursday, March 08, 2012 - 4:45 pm
I am conducting a small empirical analysis comparing CFA and Bayesian (via BUGS) approaches to estimating unidimensional IRT models. Given the relatively simple nature of my models (one dimension, 20 indicators, dichotomous scoring, sample sizes varying from 50-1,000), I am using an FIML w/ standard integration approach to estimation. As my data is simulated, I don't have any missing values. I know from previous experience and from reading through these forums the the ML approach is more efficient than the WLSMV approach, and is preferable under conditions of limited dimensionality. My questions is whether or not someone can point me to some empirical work that supports this so as I can read/verify/cite. Thank you in advance!
 Bengt O. Muthen posted on Thursday, March 08, 2012 - 5:31 pm
We have a paper discussing the performance of Bayes for IRT with binary items on this web site,

Asparouhov, T. & Muthén, B. (2010). Bayesian analysis of latent variable models using Mplus. Technical Report. Version 4.

There is also the least-squares vs ML article

Forero & Maydeu-Olivares (2009). Estimation of IRT graded response models: Limited versus full information methods. Psych. Methods, 3, 275-299.

And then there is the classic article which shows that ML isn't that much more efficient than using tetrachoric correlations (WLMSV):

Mislevy (1986). Recent developments in the factor analysis of categorical variables. Journal of Educational Statistics,. 11, 3-31.

It is very easy in Mplus to do your own simulation study, comparing Bayes, ML, and WLSMV.
 Minnik Findik posted on Friday, March 09, 2012 - 6:38 am
Dear professors,

I have a question with regards to mplus handling missing data. When I choose WLSMV and do not put "listwise deletion", how is the missing data being handled? Similar to maximum likelihood does wlsm uses all the avialble information??

Secondly, when I'm using WLSMV with missing data (and have 5 factors, and 1 observed variable), how do I know how many subjects are used in the model? And if not will be reporting the number of observations when writing a paper?

Kindest Regards.
 Bengt O. Muthen posted on Friday, March 09, 2012 - 12:26 pm
Q1. See UG, pp. 7-8.

Q2-Q3. The maximum sample size is printed in the output - report that. WLSMV uses pair-wise present data for estimating the sample correlations.
 Minnik Findik posted on Saturday, March 10, 2012 - 5:51 am
Dear Tihomir Asparouhov,

after your advice I tried to run my model as

Usevariables are
mv81 mv115 mv97 CV8 CV10 PSH PSS PSHo FAIC Act24 Adap24 Inten24 AnxT depT PSConf phy emo;
Categorical are
mv81 mv115 mv97 PSH PSS PSHo PSConf
phy emo AnxT depT CV8 CV10;


integration = monte (5000);
estimator = ml;


alient by Act24 Adap24 Inten24;
alienp by PSS PSHo PSH;
aliend by emo phy PSConf;
alienm by depT AnxT;
alienvic by mv97 mv81 mv115 CV8 CV10;
alienp with alient;
alienp with aliend;
alient with aliend;
alient on FAIC;
alienp on FAIC;
aliend on FAIC;
alienp on alienm;
alient on alienm;
aliend on alienm;
FAIC with alienm;
alienvic on alienp;
alienvic on alienm;
alienvic on FAIC;
alienvic on aliend;
alienvic on alient;

output: stand;

It starts running the analysis but it is very very slow in the MS-DOS screen and after 2 hours my computer crashes. Do you have any suggestions with regards to this??

Thanks a lo,
Kind Regards.
 Linda K. Muthen posted on Saturday, March 10, 2012 - 6:44 am
Please send the relevant files and your license number to
 Lisa M. Yarnell posted on Thursday, June 14, 2012 - 8:37 pm
Hello, I was reading that with MLR estimation, Mplus by default uses 10 sets of random starting values, run through 20 iterations, to avoid local solutions in the likelihood function. I have a couple questions:
1) These multiple runs make standard errors smaller, right?
2) With ML estimation, this "10 random starting values" process is not done, right? So, this is unique to the MLR estimation?
3) The nomenclature "robust standard errors" for MLR estimation is curious to me because it is the beta estimates that become robust, right? The standard errors become smaller, and the beta estimates become more robust to multiple runs with different local solutions, and to nuances of specific samples from a given population. Is that right? Please do correct my thinking where I am wrong.

 Linda K. Muthen posted on Friday, June 15, 2012 - 12:57 pm
1. Random starts are not used with all analyses. They are used primarily with mixture modeling. They do not make the standard errors smaller.

2. Random starts are used with ML and MLR.

3. No, it is not the parameter estimates that are robust. It is the standard errors. Everything you say is incorrect.
 Bennie posted on Monday, June 18, 2012 - 2:22 pm
Hello, I am a new researcher and have some questions as I have become confused with what I have read on this site and on other sites.

1) Is the WLSMV estimator the best option in research at the moment for categorical variables when investigating a model?

2) If the WLSMV estimator is used, is it fine to report CFI, TLI and RMSEA? As I have read that the WRMR is still a bit unreliable? So in other words is keeping with the rule of thumb of three fit indices for a model acceptable?

Thank you in advance for your replies.
 Linda K. Muthen posted on Monday, June 18, 2012 - 4:05 pm
1. Both WLSMV and ML are good estimators for categorical variables. WLSMV is probit. The ML default is logistic but probit is also available. If you have a model with many factors with categorical indicators, WLSMV is less computationally demanding. Also, if you want to include residual covariances between categorical observed variables, WLSMV is less computationally demadning. ML has better missing data handling.

2. WRMR is an experimental test statistic. I would consider all of the other test statistics.
 Aylin posted on Wednesday, June 20, 2012 - 4:12 am
Dear Linda,
In my path analysis (using WLSMV) if I include correlations between two categorical variables is this correlation a tetrachoric correlation? and if I look at the correlations between two continuous variables is it pearson r correlations??
Thank you.
 Linda K. Muthen posted on Wednesday, June 20, 2012 - 10:36 am
If you are looking at sample statistics, you will find either tetrachoric or polychoric corelations for categorical variables and Pearson correlations for continuous variables. If you are looking in the results section of the output, it will be the same for categorical but covariances for continuous variables.
 Aylin posted on Wednesday, June 20, 2012 - 10:55 am
I was referring to the results section of the output. I am not sure what do you mean by "it will be same for categorical". So will it be tetrachoric correlations??
thank you
 Aylin posted on Wednesday, June 20, 2012 - 12:17 pm
And another quick question, then what are the correlations (in the results section of the output) of latent variables with WLSMV estimator? Are they covariances and not Pearson R correlations?
 Linda K. Muthen posted on Wednesday, June 20, 2012 - 12:39 pm

They are covariances.
 Aylin posted on Wednesday, June 20, 2012 - 1:58 pm
Thank you very much Linda.
Is there a way to estimate the Pearson r correlations between latent variables? Or am I completely on the wrong page?
 Linda K. Muthen posted on Wednesday, June 20, 2012 - 6:30 pm
Ask for TECH4 in the OUTPUT command.
 Jan Zirk posted on Friday, July 20, 2012 - 6:51 am
Dear Linda or Bengt,

I have a mediation model with a group mediator (b).

Model 1: c on b; b on a; c on a;
Pearson correlation matrix shows strong negative correlation c with a and strong negative correlation/beta for c with b;
MLR shows that 'c on a' in Model 1 is highly not sign. Bayesian estimation and WLSMV show that 'c on a' is significant and the estimate is unexpectedly positive.
When the group mediator is taken out (Model 2: c on a), 'c on a' becomes negatively significant under MLR, WLSMV and Bayesian estimation.
There can be some conceptual/phenomenological explanation of this paradoxically positive direction of the relationship in the mediatio model, but I wonder what is the possible common feature of Bayes and WLSMV which make them able to detect that in Model 1 paradoxically 'c on a' is highly positively sign while MLR & ML fail to do this?
 Bengt O. Muthen posted on Friday, July 20, 2012 - 8:54 pm
You mention that your mediator b is a "group mediator" by which I assume it is binary. ML estimation uses the observed binary mediator b as the predictor of c. In contrast, WLSMV uses the continuous latent response variable b* behind b as the predictor. The default for Bayes does the same as WLSMV, but Bayes can also use the observed mediator as ML does (using the mediator= option of the Analysis command).

As argued in the paper below, the standard approach of estimating an indirect effect as a product of two coefficients is only appropriate using the b* approach. If b itself is the substantively motivated mediator you need to instead approach the indirect effect estimation as is done in the paper based on "causally defined" effects.

Muthén, B. (2011). Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus.

This paper is on our web site under Papers, Mediational Modeling.
 Jan Zirk posted on Saturday, July 21, 2012 - 4:06 am
Dear Bengt,
Thank you so much for your generous and precise help. Indeed, my "group mediator" is binary.

Best wishes,

 Jan Zirk posted on Saturday, July 21, 2012 - 4:24 am
"the standard approach of estimating an indirect effect as a product of two coefficients is only appropriate using the b* approach."

Does this mean that the MLR coefficients are completely uninterpretable and the MLR information criteria can not be used as evidence for model preference over the alternative?
 Bengt O. Muthen posted on Saturday, July 21, 2012 - 9:05 am
The coefficients for a->b and from b->c are interpretable but the indirect effect cannot be said to be the product of those 2 coefficients.
 Jan Zirk posted on Saturday, July 21, 2012 - 9:16 am
Oh, I see; thank you very much for help.

 Trang Q. Nguyen posted on Thursday, July 11, 2013 - 4:14 pm
Would you please help me in assessing model fit for an SEM model with categorical data using MLR? I am not familiar with MLR, and appreciate all guidance.

The model is below, where outcome is binary and pain1-3 are 4-category ordinal variables. I am using MLR because I am interested in estimating an odds ratio for outcome associated with the pain factor.

NAMES = id sex age outcome pain1 pain2 pain3;
USEVAR = sex age outcome pain1 pain2 pain3;
CATEGORICAL = outcome pain1 pain2 pain3;


pain by pain1* pain2 pain3;

outcome ON pain sex age;

For model fit, I got the following. How do I use this to assess if this model fits adequately to the data? Thanks much!


Number of Free Parameters 16


H0 Value -1784.887
H0 Scaling Correction Factor 1.7027
for MLR

Information Criteria

Akaike (AIC) 3601.774
Bayesian (BIC) 3668.884
Sample-Size Adjusted BIC 3618.100
(n* = (n + 2) / 24)
 Bengt O. Muthen posted on Thursday, July 11, 2013 - 4:53 pm
This fit information is to be used for comparisons with other competing models.
Your model states that you have no direct effects from the pain factor indicators to the outcome. So the model can be tested by adding one such direct effect at a time.

For absolute fit to the data you can request TECH10 in the OUTPUT. You can also estimate this with WLSMV or Bayes to get fit statistics, although that would bring you to using probit, not logit.
 Anonymous posted on Friday, September 06, 2013 - 5:19 pm

I am running a mediational path model (no SEM) that has both continuous and categorical observed variables. I have a continuous variable that is skewed (it is not an outcome) and I am wondering what estimator would be best to use with this type of model. I had previously run the model using WLSMV because I understand that it will handle continuous and categorical outcomes, however I am unsure of whether WLSMV is robust to non-normal data. If not, is it appropriate to use MLR with this model?
 Linda K. Muthen posted on Saturday, September 07, 2013 - 6:23 am
WLSMV is not robust to non-normal continuous variables. If your continuous variables are non-normal, I would suggest using MLR which is robust to non-normality and can also handle a combination of continuous and categorical observed variables. One issue with categorical variables and maximum likelihood estimation is that each factor with categorical factor indicators requires one dimension of integration and you don't want a model with too many dimensions of integration.
 David Bayliss posted on Monday, February 10, 2014 - 1:39 am

I have a continuous dependent variable from complex survey data which exhibits kurtosis and so is non-normal. I am running regressions to explore the impact of the kurtosis on the residuals. I have read Yuan & Bentler (2000) and Muthén & Asparouhov (2002) which show the MLR estimator to perform well when the dependent variable has kurtosis of a similar amount to the variable I have. My question is, should I expect regression residuals to be more normally distributed after having estimated a regression using MLR compared to a non-robust estimator, or does it not work this way?

Kind regards.
 Bengt O. Muthen posted on Monday, February 10, 2014 - 11:31 am
I don't think the residuals will be different because ML and MLR do not differ in terms of estimates, only in terms of SEs.
 Eivind Ystrøm posted on Wednesday, February 12, 2014 - 1:25 am
Why is maximum likelihood is preferable to weighted least squares when comparing the fit of nested measurement models with categorical indicators? Do you have a reference on this?
 Bengt O. Muthen posted on Wednesday, February 12, 2014 - 3:01 pm
I don't know that ML is preferable in this case.
 thanoon younis posted on Thursday, February 13, 2014 - 7:03 pm
dear dr. Muthen
i want to modified WLSMV and i want to change link function to another question can i change the link function in WLSMV.
 Bengt O. Muthen posted on Friday, February 14, 2014 - 10:46 am
WLSMV only allows the probit link function. This is what makes WLSMV modeling very general because you have access to residual correlation parameters which you don't as easily with logit link.
 thanoon younis posted on Wednesday, February 19, 2014 - 6:49 pm
hi dr. MUTHEN
i want to ask you regarding WLSMV and WLS can i use these methods with categorical data without outliers.

thanks alot
 Linda K. Muthen posted on Thursday, February 20, 2014 - 10:15 am
 thanoon younis posted on Sunday, March 16, 2014 - 8:10 pm
i want to know what you means in adjusted mean and variance in WLSMV mehod? how you adjust mean and variance?
thanks in advance.
 Linda K. Muthen posted on Monday, March 17, 2014 - 2:45 pm
See the following paper which is available on the website:

Muthén, B., du Toit, S.H.C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes.
 Mariah Mantsun Cheng posted on Wednesday, April 23, 2014 - 9:21 am
Dear Bengt & Linda,
I am working on an exploratory factor analysis on 9 variables among which 3 are categorical (yes/no) and the other 6 are continuous, controlling for complex survey design. I have tried using the default WLSMV estimator and the MLR estimator. However the results are quite different.
While I prefer the MLR results, in the MPlus output there are warnings of saddle points occurred during the estimation processes.
1. Given the warnings, should I trust the MLR results? Since Mplus seems to have done some adjustments, does that mean the problems have been fixed and results reliable?
I have re-run the programs by increasing the number of integration points, or decreasing the value of the MCONVERGENCE. But either way, I still got the saddle point warnings. Actually, with increased integration points, I got 0 solutions for my estimates.
2. What would you suggest in order to resolve this saddle point problem? Can I change the starting values easily in the estimation?
3. Should I expect different results by using MLSMV vs. MLR estimator? I do have some missing data in these variables (1-4%).
Thank you for your advices in advance!
 Linda K. Muthen posted on Wednesday, April 23, 2014 - 3:21 pm
Results can be different because WLSMV uses a probit model and MLR uses an logistic model as the default. Missing data handling is also different between the two estimators.

Please end the MLR output and your license number to

MLMV uses listwise deletion and is not available for categorical variables.
 Rachael Robnett posted on Friday, May 30, 2014 - 3:00 pm
Dear Drs. Muthen:

I used Mplus to test a mediational path model. ML was my original estimation method, but a reviewer suggested that this was inappropriate because my variables are technically ordinal (created from participants' responses to 5-pt Likert scales).

Here is my question: Is MLR appropriate for testing mediation models? An exchange above implied that this might not be the case, but I wanted to verify.

Many thanks!
 Linda K. Muthen posted on Friday, May 30, 2014 - 3:35 pm
Yes, MLR is appropriate. The CATEGORICAL option can be used with maximum likelihood estimation.
 Rachael Robnett posted on Friday, May 30, 2014 - 4:39 pm
Thank you for your response. That is great news.

Is the significance test of the indirect path interpreted in the same way for ML vs. MLR? I ask because I came across the following quote above, which was made in reference to MLR:

"The coefficients for a->b and from b->c are interpretable but the indirect effect cannot be said to be the product of those 2 coefficients." [from B. Muthen]

Thank you again. This is immensely helpful.
 Linda K. Muthen posted on Friday, May 30, 2014 - 4:48 pm
The comment is not related to whether you use ML or MLR. The difference between ML and MLR is that the standard errors are different.

The comment relates to the fact that not all indirect effects can be computed using the product of the two regression coefficients. See the following paper on the website for further information:

Muthen, B. & Asparouhov T. (2014). Causal effects in mediation modeling: An introduction with applications to latent variables. Forthcoming in Structural Equation Modeling.
 Rachael Robnett posted on Friday, May 30, 2014 - 4:55 pm
Noted. Thank you for bearing with my confusion and for the reference.
 Jenny Häufle posted on Friday, June 27, 2014 - 5:00 am
Dear Professor/s,

I want to analyze a multiple indicator, quadratic growth model with four measurement time points. The growth parameters are predicted by another latent variable that also has multiple indicators. My indicators are all categorical and I have substantial missing data. I don't believe that my data meet MCAR assumtions but I have reason to believe that the data are MAR. I do also have some observed variables that might be able to predict missingness.
As I understand MLR is better in handling missing data. But WLSMV is less time consuming with many dimensions and allows MAR when there are covariates in the model. Which estimator would you recommend and do you have any citation I can use to justify this decision?

 Linda K. Muthen posted on Friday, June 27, 2014 - 11:07 am
I would use maximum likelihood because it is unlikely that the missingness can be well explained by only the predictors which is done in WLSMV. If you have a lot of missing data, this can affect your results. I don't know what you mean by substantial.
 Jenny H posted on Monday, June 30, 2014 - 2:24 am
Thank you so much Linda for your quick response!

I have minor missingness due to item non-response but I do have quit a lot missingness due to dropout (arount 30 % between the measurement time points). I have a prison sample and people mainly drop out due to release from prison. I do have a variable asking the people how close they are to release, so I can incorporate this as an auxiliary variable.
Since my data are non-normal, I will use MLR. However, with MLR I don't get fit statistics. Is there any way to obtain some fit information that I can use when writing this up?
 Linda K. Muthen posted on Monday, June 30, 2014 - 6:38 am
There are no absolute fit statistics with maximum likelihood and categorical data. Nested models can be compared using -2 times the loglikelihood difference which is distributed as chi-square. Non-nested models with the same dependent variables can be compared using BIC.
 Jenny H posted on Wednesday, July 02, 2014 - 3:32 am
Dear Linda,

thanks for your helpful advise!
I have a follow-up question: I tried to incorporate auxiliary variables as missing data correlates into my multiple indicator growth model. However, I get an error message that the "Analysis with categorical variables is not available with the 'm' specifier in the AUXILIARY option."
Is it right that I can't use missing data correlates with categorical indicators?

Since I have observed variables that predict missingness is there any way to incorporate these variables?
Would it be better in that case to treat the indicators (with 4 categories) as continuous and use auxiliary variables?
 Bengt O. Muthen posted on Wednesday, July 02, 2014 - 10:15 am
Yes, the Auxiliary m option has not been developed for the categorical case. It is possible, although cumbersome, for an advanced user to do this by hand by following our short course handouts on the topic. Note, however, that Auxiliary m additions in my experience seldom change the results very much.

You can include the missing-data predicting variables in the model as covariates if those variables also play a substantive role in the model.

If your 4-category variables don't have strong floor or ceiling effects, approximating them as continuous might not be bad, but it seems a pity to have to change the model in this way.
 Jenny H posted on Friday, July 04, 2014 - 1:46 am
Dear Bengt, dear Linda,

I had a look at the handouts on the topic of missing data analysis. Am I right in assuming that I can introduce the missing data correlates into my model by allowing them to correlate with all of my categorical indicators and with each other? Is that what the auxiliary (m) option basically does? I don't think that the variables play a substantive role in the model but I at least want to check once and see if results change.

Thank you so much for your advise!
 Bengt O. Muthen posted on Friday, July 04, 2014 - 8:51 am
Yes; that's it. But there are details to consider in implementing it (lots behind the scenes in Aux (m)). And if you use ML, correlating categorical outcomes with other variables is not straightforward. Having them as covariates is far easier, but may not be right from a substantive point of view.
 ri ri  posted on Wednesday, December 03, 2014 - 4:02 pm
I used MLR to conduct a BIC difference test, since WLSMV can not be applied in BIC (I have binary dependent variables). As I used WLSMV in some of my hypothesis testings and used MLR for BIC difference test (in order to compare models). Would it be a Problem that I used both WLSMV and MLR, though the purposes are different?

Could you recommend 1-2 papers that compare MLR and WLSMV in testing binary or categorical variables? What is the main difference in Terms of results?

 Bengt O. Muthen posted on Wednesday, December 03, 2014 - 4:26 pm
Q1. As long as you state which estimator the various results come from, you are ok. But you can get both BIC/DIC and chi-square testing using PPP if you choose Estimator=Bayes.

Q2. See the paper on our website:

Barendse, M.T., Oort, F.J., & Timmerman, M.E. (2014). Using exploratory factor analysis to determine the dimensionality of discrete responses. Structural Equation Modeling: A Multidisciplinary Journal, 00: 1-15.
 Helen Skerman posted on Thursday, February 05, 2015 - 4:46 pm
I am wanting to cite this paper:
Muthén, B., du Toit, S.H.C. & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Accepted for publication in Psychometrika.

Was this paper published? Could you please advise appropriate citation.
 Linda K. Muthen posted on Thursday, February 05, 2015 - 6:52 pm
Muthén, B., du Toit, S.H.C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Unpublished technical report.
 Cheng posted on Wednesday, April 22, 2015 - 2:57 am
Dear Linda,
Is there any requirement of sample size for using WLSMV estimator in CFA?
 Joost de Moor posted on Thursday, May 07, 2015 - 3:11 am
Dear Professors,

I am using the WLSMV estimator for a structural model with binary outcome variables. However, I want to present my measurement model first. This, however, does not have any categorical variables in it, which causes MPLUS to report the following error message:

*** WARNING in ANALYSIS command
Estimator WLSMV is not available for analysis with all continuous variables.
Default estimator will be used.

In order to compare the measurement model and the structural model, I would prefer to use an estimator with a probit link function in both. Could you please tell me what other estimator than WLSMV I could use for the measurement model without categorical variables so that the results of both models remain comparable?

 Linda K. Muthen posted on Thursday, May 07, 2015 - 5:49 am
Maximum likelihood.
 LIDYANE CAMELO posted on Wednesday, June 17, 2015 - 1:19 pm
Dear all,

I am working in a SEM model with categorical and continuous variables using WLSMV. Two of my mediators are continuos variables and non-normally distributed. I saw in some papers that WLSMV is robust in case of non-normality, especially in large samples (I am working with N=6000). However, I read some comments here saying that WLSMV is not robust to non-normality in continuous variable. Thus, I have tried to perform my model using MLR (as you suggest in comments above), but the model did not converge. Using the WLSMV, I have performed my model (you can see my model below) with the non-normal variables, and after I have performed the same model with the non-normal variables natural log-transformed, and the results were very different. Now I do not know which model is better to report. Which one do you believe that is better?

Do you have some references about the fact that WLSMV is not robust in case of non-normality?

My model:
F1 BY U1-U3;
Y1 ON Y2 Y3;
Y4 ON F1 Y1 Y2 Y3;
F1 ON Y1 Y2 Y3;
Y5 ON F1 Y1 Y2 Y3 Y4;
U4 ON F1 Y1 Y2 Y3 Y4 Y5;

Thank you in advance
 Bengt O. Muthen posted on Wednesday, June 17, 2015 - 6:06 pm
WLSMV is not robust to non-normal continuous variables.

I recommend using MLR and if you have problems, send to support along with your license number.

I would not transform the variables; it is often better if statistics adjust to the data than the other way around.
 LIDYANE CAMELO posted on Thursday, June 18, 2015 - 5:59 am
I believe that the problem with MLR, it is because I have mediators variables continuous and binary. I have read before that MLR is not good to deal with binary mediators. In these case, do you recommend the use of WLSMV with transformation of non-normal continuous variable?
 Bengt O. Muthen posted on Thursday, June 18, 2015 - 12:02 pm
I see. Well then, try Bayes where you get the choice to treat the binary mediators as latent mediators (continuous, latent response variables behind the binary observed variables).
 Alissa Mahler posted on Friday, November 13, 2015 - 1:49 pm

I am running a cross-lag model with two continuous variables at 3 time-points. One of my variables is highly skewed, so I used the censoring option, and to get fit statistics I am using the WLSMV estimator. I am not familiar with this estimator and I had a few questions:

(1) The sample statistics provided using the WLSMV estimator do not appear to map onto the means of my actual variables. When I run basic descriptives in Mplus they are estimated correctly, so I don't think it is an issue of the data being read incorrectly. Are sample statistics provided using the WLSMV estimator not the means of my variables?

(2) Based on previous discussions, when using WLSMV, standard errors and p-values are not generated for the standardized results. Does this mean the standardized results cannot be reported?

Thank you!
 Bengt O. Muthen posted on Friday, November 13, 2015 - 5:38 pm
(1) With x's in the model the sample statistics are different - they are regression estimates. See Muthen (1984) under Papers, SEM.

(2) Are you using Mplus Version 7.4?
 Alissa Mahler posted on Saturday, November 14, 2015 - 9:01 am
Hi Bengt,

Thanks for your quick reply. I will check out that paper. To answer your question, I am using version 5.1. Unfortunately I am using my school's license, and they have not updated the software. If I cannot get access to version 7.4, how do you recommend reporting the standardized results?

Thanks again!
 Bengt O. Muthen posted on Saturday, November 14, 2015 - 2:53 pm
You would then have to standardize yourself, expressing the standardized coefficients in Model Constraint using parameter labels that you give in the Model command.
 Margarita  posted on Wednesday, April 06, 2016 - 7:42 am
Dear Dr. Muthén,

I have a 2-factor CFA model with categorical indicators, but my data have missing values and some items present high skewness and kurtosis. I understand that in such cases MLR or Bayes would be a sufficient alternative to WLSMV because they use full-information. Would MLR and Bayes be a good way to go? My only concern is that Bayes does not provide fit indices and can be very conservative when it comes to model fit.

As always, thank you for your help.
 Bengt O. Muthen posted on Wednesday, April 06, 2016 - 12:35 pm
Skewed categorical variables is not a problem, but missing data can be for WLSMV. ML and Bayes handle missing data better. Neither ML nor Bayes provide much fit information with categorical. ML does give bivariate fit info using TECH10.
 Margarita  posted on Wednesday, April 06, 2016 - 2:08 pm
Thank you for your response. One last question. Would it be wrong to treat categorical indicators as continuous?
 Linda K. Muthen posted on Wednesday, April 06, 2016 - 2:36 pm
Categorical items that have floor or ceiling effects, low or high categories with a pilng up of observations, should be treated as categorical.
 Margarita  posted on Thursday, April 07, 2016 - 3:34 am
So, for categorical items with floor effects, and 5-6% missing data, you would suggest MLR/Bayes and treating them as categorical?
 Linda K. Muthen posted on Thursday, April 07, 2016 - 8:53 am
That is not a lot of missing data so WLSMV may be fine. You can do both MLR and WLSMV. If there's a big difference, I would use MLR because it has better handling of missing data.
 Will Vincent posted on Sunday, June 19, 2016 - 2:45 pm

I am estimating two models using MLR estimation as follows:

Model 1
Bivariate correlations between continuous and categorical observed variables and continuous latent variables where all variables are treated as continuous as required to estimate correlations under maximum likelihood estimation, and

Model 2
Exposure -> Mediator -> Outcome, where the model is a logit model in which exposure variables are continuous and categorical observed variables and continuous latent variables, mediator variables are continuous latent variables, and the outcome variable is a dichotomous variable.

In model 1, categorical variables are not specified (i.e., CATEGORICAL ARE....) as such, which I am thinking is yielding Pearson or biserial correlations. In model 2, the dichotomous outcome is specified as categorical. Can the AIC and BIC fit statistics between these two models be compared?

Thank you for your assistance.
 Bengt O. Muthen posted on Monday, June 20, 2016 - 9:05 am
 Peter Taylor posted on Thursday, April 13, 2017 - 5:47 am
Hi, could you recommend some introductory reading on the use of ML with categorical variables/indicators? There seems to be a lot on WLSMV but less on the ML approach to this kind of data. Thanks
 Bengt O. Muthen posted on Thursday, April 13, 2017 - 4:07 pm
There is the whole Item Response Theory literature. We also discuss some of it in our Hopkins short course videos. And a quick summary is given in the FAQ:

Estimator choices with categorical outcomes
 Peter Taylor posted on Wednesday, April 19, 2017 - 12:01 am
Thanks for your help
 Cheng posted on Tuesday, May 30, 2017 - 10:55 pm
May I confirm whether I should use WLSMV estimator if my SEM model consists of

1) endogenous variable (latent) consists of observed variables which are categorical/dichotomous

2) endogenous variable (latent) consist of observed variables which are count number

3) endogenous variable (latent) consist of observed variable which are numerical (interval)

4) the exogenous variables consist of numerical and categorical variables.
 Linda K. Muthen posted on Wednesday, May 31, 2017 - 9:49 am
You will need to use MLR. WLSMV cannot be used with a count outcomes.
 Cheng posted on Wednesday, May 31, 2017 - 9:13 pm
If I use MLR, do I need to exclude the latent (endogenous) with observed dichotomous variables from the SEM model? My understanding is MLR not suitable for dichotomous dependent variable. Please correct me if I am wrong. Thank you very much.
 Bengt O. Muthen posted on Thursday, June 01, 2017 - 6:53 pm
You are wrong - MLR does not imply that variables have to be continuous. MLR is fine for categorical, cunt and censored variables as well.
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